Let $S$ be a region in the plane with area 10.  When we apply the matrix
\[\begin{pmatrix} 2 & 1 \\ 7 & -3 \end{pmatrix}\]to $S,$ we obtain the region $S'.$  Find the area of $S'.$
Explanation: Note that
\[\begin{vmatrix} 2 & 1 \\ 7 & -3 \end{vmatrix} = (2)(-3) - (1)(7) = -13,\]so the matrix scales the area of any region by a factor of $|-13| = 13.$  In particular, the area of $S'$ is $13 \cdot 10 = \boxed{130}.$